waitedavid137
Honorable
So let's start by writing down the Robertson-Walker solution for a universe of uniform matter distribution.
where
is a function of
.
And
and
They constructed this as a model in such a way that the matter distribution around an observer would look the same from his vantage point, no matter where he was located in that universe. Despite clumping of matter as we observe as galaxies etc... it is a small variation from the statistical behavior of the overall matter, including especially cosmic microwave background radiation, so they expected this to be a good model for the average behavior of the universe.
Calculation from that spacetime yields the following Einstein tensor elements,
all other elements are zero.
Now Einstein's field equations are
where
So the first equation I wrote for the
element calculated for the spacetime's Einstein tensor given that element for his field equations results in
which can be rewritten as
This is one of the two Friedmann equations, where a lot of folk define
Next take the second equation I wrote for the
element calculated for the spacetime's Einstein tensor given that element for his field equations results in
Define
for pressure as
Now insert this and the result for
to arrive at
This can be rewritten
which is the other of Friedmann's equations.
where
And
and
They constructed this as a model in such a way that the matter distribution around an observer would look the same from his vantage point, no matter where he was located in that universe. Despite clumping of matter as we observe as galaxies etc... it is a small variation from the statistical behavior of the overall matter, including especially cosmic microwave background radiation, so they expected this to be a good model for the average behavior of the universe.
Calculation from that spacetime yields the following Einstein tensor elements,
all other elements are zero.
Now Einstein's field equations are
where
So the first equation I wrote for the
which can be rewritten as
This is one of the two Friedmann equations, where a lot of folk define
Next take the second equation I wrote for the
Define
Now insert this and the result for
This can be rewritten
which is the other of Friedmann's equations.
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